Abstract
If \(Z_t = \rho _t e^{i\theta _t }\)is a continuous, complex-valued martingale, is it possible that, with positive probability, both ρt and ϑt tend to infinity when t→∞? If Z is a conformal martingale, the answer is clearly no (for both Log ρt and ϑ t are local martingales too). But if conformality is not required, such a behavior is possible. This note gives an example of a planar spiral curve σ and a continuous martingale that never hits σ but still has a non-zero probability of escaping to infinity.
Research supported in part by N.S.F. grant MCS80-02535.
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References
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© 1991 Springer-Verlag
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Dubins, L.E., Emery, M., Yor, M. (1991). A continuous martingale in the plane that may spiral away to infinity. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXV. Lecture Notes in Mathematics, vol 1485. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0100862
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DOI: https://doi.org/10.1007/BFb0100862
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